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Functional Dependencies (FD) Let X be a set of attributes of a relation R Let A be a single attribute of R X A holds for R if:
whenever two tuples of R agree on all the attributes of X,
then they must also agree on the attribute A. We say X “uniquely determines” A in R
Example Customer(Name, Addr, SodaLiked, Manf,
FavSoda), with name identifying a unique person
Lots of redundancy here…Name Addr SodaLiked Manf FavSoda
Janeway Voyager Pepsi PepsiCo Coke
Janeway Voyager Sprite CocaCola Coke
Spock Enterprise Pepsi PepsiCo Coke
FDs from Data Does Name Addr?
Name Addr SodaLiked Manf FavSoda
Janeway Voyager Pepsi PepsiCo Coke
Janeway Voyager Sprite CocaCola Coke
Spock Enterprise Pepsi PepsiCo Coke
FDs from Data Does Name Addr? Yes, since we assumed unique names
Name Addr SodaLiked Manf FavSoda
Janeway Voyager Pepsi PepsiCo Coke
Janeway Voyager Sprite CocaCola Coke
Spock Enterprise Pepsi PepsiCo Coke
FDs from Data Does Name FavSoda?
Name Addr SodaLiked Manf FavSoda
Janeway Voyager Pepsi PepsiCo Coke
Janeway Voyager Sprite CocaCola Coke
Spock Enterprise Pepsi PepsiCo Coke
FDs from Data Does Name FavSoda? Yes, we want just one favorite per person
Name Addr SodaLiked Manf FavSoda
Janeway Voyager Pepsi PepsiCo Coke
Janeway Voyager Sprite CocaCola Coke
Spock Enterprise Pepsi PepsiCo Coke
FDs from Data Does SodaLiked Manf?
Name Addr SodaLiked Manf FavSoda
Janeway Voyager Pepsi PepsiCo Coke
Janeway Voyager Sprite CocaCola Coke
Spock Enterprise Pepsi PepsiCo Coke
FDs from Data Does SodaLiked Manf? Yes, since each soda has just one manf.
Name Addr SodaLiked Manf FavSoda
Janeway Voyager Pepsi PepsiCo Coke
Janeway Voyager Sprite CocaCola Coke
Spock Enterprise Pepsi PepsiCo Coke
FDs from Data Does FavSoda Name?
Name Addr SodaLiked Manf FavSoda
Janeway Voyager Pepsi PepsiCo Coke
Janeway Voyager Sprite CocaCola Coke
Spock Enterprise Pepsi PepsiCo Coke
FDs from Data Does FavSoda Name? No, two people might have the same favorite
Name Addr SodaLiked Manf FavSoda
Janeway Voyager Pepsi PepsiCo Coke
Janeway Voyager Sprite CocaCola Coke
Spock Enterprise Pepsi PepsiCo Coke
FDs from ER Diagrams From entity sets
(Key of entity set) other attributes of entity set From many-one relationship
(Key of “many” set) attributes of “one” set
Notation Shorthand Technically FDs go from sets to single
attributes { Name } Addr { Name } FavSoda
Often just combine to write: Name Addr, FavSoda
Usually omit set braces on left side also: Restaurant, Soda Price
Keys Revisited Let K be a set of attributes of a relation R K is a super key for R if:
For all attributes A in R, K A K is a key for R if:
No proper subset of K is a super key for R An attribute B is a prime attribute of R if:
B is an element of some key of R
Example What is the key here? What are the prime attributes?
Customer(Name, Addr, SodaLiked, Manf, FavSoda)
Two Ways to Find Keys Guess a superkey K:
Show that K A for all attributes A Show that no subset of K is a superkey
Find all functional dependencies Check all possible keys
Redundancy Leads to Anomalies Update anomaly: one occurrence of a fact is
changed, but not all occurrences Deletion anomaly: valid fact is lost when a
tuple is deleted
Example
Name Addr SodaLiked Manf FavSoda
Janeway Voyager Pepsi PepsiCo Coke
Janeway Voyager Sprite CocaCola Coke
Spock Enterprise Pepsi PepsiCo Coke
Redundant with first row since Name Addr, FavSoda
Redundant with first row since SodaLiked Manf
Third Normal Form A relation R is in Third Normal Form (3NF)
if whenever X A is a nontrivial functional dependency that holds in R, then either: X is a superkey for R, or A is a prime attribute of R
Normalization Algorithm To normalize a relation R:
Find the functional dependencies for R Check that whether each FD satisfies 3NF
If so, we’re done and R is normalized
Otherwise let X A be an FD that violates 3NF Find the closure of X in R, denoted X+
Split R into new relations (R - X+ + X) and X+
Repeat algorithm for each new relation
Step 2: Check for 3NF Violations A relation R is in Third Normal Form (3NF)
if whenever X A is a nontrivial functional dependency that holds in R, then either: X is a superkey for R, or A is a prime attribute of R
Step 3: Pick a Violating FD, Find Closure
For X A the closure of X, denoted X+, is: The set of all attributes that can be reached from
any subset of X by following any FDs Or, just follow the arrows